1 Mode-suppression A simple and provably stable chunk sharing algorithm for P2P networks Srinivas Shakkottai Texas A&M University Joint work with Vamseedhar Reddyvari (TAMU) and Parimal Parag (IISc)
Peer to Peer Network 2 v P2P network offers many advantages over Client-Server approach • Scalability • Decrease the cost of distribution • Build robustness v 30% of P2P traffic in Asia-Pacific region in 2016 v BitTorrent is the popular P2P application used for file sharing v Spotify uses a combination of client server and P2P for music streaming and downloads v Microsoft is using P2P for distributing Windows 10 updates
P2P File sharing: System Model 3 § File is divided into 𝑛 chunks … § New peers enter the system with no chunks 1,2, …… .. ,m § Arrival process is Poisson( 𝜇 ) § Peer leaves the system as soon as it receives all the chunks 𝜈 v There always exists a seed that posseses 𝜇 all the chunks v Seed contacts a peer according to a Poisson (𝑉) contact process v Every peer contacts another peer(s) 𝑉 according to a Poisson( 𝜈 ) contact process v Chunks are transmitted according to given Seed … chunk selection policy 1,2, …… .. ,m
Hajek’s – (In) Stability Result 4 § The system is said to be stable for a given 𝜇 if the Markov Chain is positive recurrent § In general if a peer selection is random then any "work conserving” policy will be unstable if 𝜇>𝑉 Rarest First Policy: Find the list of useful chunks and among them select the chunk with least marginal chunk frequency § Mendes, Towsley et.al observed that P2P networks following the BitTorrent protocol show unstable behavior due to the formation of large One clubs.
Why Unstable? Missing Chunk Syndrome 5 One Club: Group of peers which have all the chunks except one particular chunk Infected Peers: Peers that posses the chunk that one club peers are missing § An entering peer is likely to join the One club § If a One club peer samples an infected peer, it will leave the system: the infected peers will not grow and One club keeps growing. § System becomes unstable One Club New Peer Normal Young Peers Seed Infected
Missing Chunk Syndrome in Simulations 6 One Club Formation
Related Work 7 https://www.sandvine.com/trends/global-internet-phenomena/ , 2016. [1] [3] B. Hajek and J. Zhu, “The missing piece syndrome in peer-to-peer communication,” in Information Theory Proceedings (ISIT), 2010 IEEE International Symposium on . IEEE, 2010, pp. 1748–1752. sharing systems,” Queueing Systems , vol. 67, no. 3, pp. 183–206, 2011. [4] D. X. Mendes, E. d. S. e Silva, D. Menasche, R. Leao, and D. Towsley, [12] B. Oguz, V. Anantharam, and I. Norros, “Stable distributed p2p pro- [2] tocols based on random peer sampling,” IEEE/ACM Transactions on Networking (TON) , vol. 23, no. 5, pp. 1444–1456, 2015. , vol. 23, no. 5, pp. 1444–1456, 2015. [13] O. Bilgen and A. Wagner, “A new stable peer-to-peer protocol with non- [3] [13] O. Bilgen and A. Wagner, “A new stable peer-to-peer protocol with non- persistent peers,” in Proceedings of INFOCOM , 2017, pp. 1783–1790. 4713, 2012. [14] L. Massouli´ e and M. Vojnovic, “Coupon replication systems,” [10] H. Reittu, “A stable random-contact algorithm for peer-to-peer file [4] sharing.” in IWSOS . Springer, 2009, pp. 185–192. sharing.” in IWSOS . Springer, 2009, pp. 185–192. [11] I. Norros, H. Reittu, and T. Eirola, “On the stability of two-chunk file- [11] I. Norros, H. Reittu, and T. Eirola, “On the stability of two-chunk file- [5] sharing systems,” Queueing Systems , vol. 67, no. 3, pp. 183–206, 2011. International Symposium on . IEEE, 2010, pp. 1748–1752. [12] B. Oguz, V. Anantharam, and I. Norros, “Stable distributed p2p pro- [4] D. X. Mendes, E. d. S. e Silva, D. Menasche, R. Leao, and D. Towsley, [6] “An experimental reality check on the scaling laws of swarming sys- tems,” in Proceedings of INFOCOM , 2017, pp. 1647–1655. [5] D. Qiu and R. Srikant, “Modeling and performance analysis of
Group Suppression 8 § A peer from the largest group should not give a chunk to any peer possessing fewer chunks than itself § A seed should give chunks only to the most deprived peers § The idea behind the Group Suppression is to avoid the growth of One club peers § In particular the peers in One club will not transfer chunks to new young peers if One club is large Group Suppression is stable for 𝜇 >0 if 𝑛 =2 § Has good sojourn time in general (although variable). Sojourn Time : Sojourn time is the amount of time a peer spends in the system before leaving the system by receiving all the chunks
System Model 9 § 𝑌 S (𝑢) denote number of peers with chunk profile 𝑇 § State of the System at time 𝑢 is denoted by 𝑌(𝑢) and is a vector of length 2 m −1 elements § The element at index 𝑗 is the number of peers with chunk profile corresponding to 𝑗 . § 𝑌 ( 𝑢 )[ 𝑗 ]= X 𝑇 ( 𝑢 ) where <S> = i § Total number of peers in the system is |𝑌(𝑢)|=∑ 𝑡⊂[𝑛] 𝑌 S (𝑢) 2= <0,1,0> 1= <0,0,1> 𝑌(𝑢) =(0,1,2,0,0, 0, 1) 𝑛 =3 2= <0,1,0> 6= <1,1,0>
Mode Suppression 10 Suppress chunks which are in the mode except when all the indices are in the mode 2 4 1 3 4 2 2 2 2 2 ℳ (𝑦) ={2,5} ℳ (𝑦) ={1,2,3,4,5} D (𝑦) ={2,5} D (𝑦) = 𝜚 § In the first case if a peer with ( 0,0,0,0,0 ) 𝑛𝑓𝑓𝑢𝑡 (0,1,0,0,1) then no chunk transferred
Mode Suppression – Q Matrix 11 § Let 𝐵 ( 𝑦 , 𝐶 , 𝑇 ) be the set of allowed chunks from 𝐶 to 𝑇 , then A ( x, B, S ) = B \ ( S ∪ D ( x )) ∀ S : j / ∈ S ; j / ∈ D ( x ) , Q ( x, T S,j ( x )) = x S U x T ⇣ ⌘ X | A ( x, [ m ] , S ) | + µ | x | | A ( x, T, S ) | T : j ∈ T Q ( x, x + e φ ) = λ Chunk from other Peers Chunk from Seed 𝜈 𝜇 Where, T S,j ( x ) = x − e <S> if S ∪ j = [ m ] 𝑉 = x − e <S> + e <S [ j> o.w … Seed 1,2, …… .. ,m
Stability of Mode Suppression | {z } | {z } 12 Theorem 1 The stability region of the Mode Suppression Policy is � > 0 if m ≥ 2 , U > 0 and µ > 0 . § To prove the stability we need to prove that the Markov Chain is positive recurrent § Proving positive recurrence directly is difficult in this case § So, we employ Foster-Lyapunov criteria and come up with a Lyapunov function for which the drift is negative (Foster-Lyapunov Criteria:) Suppose X ( t ) is irreducible and if there exists a function V : S → R + such that 1. P y 6 = x Q ( x, y )( V ( y ) − V ( x )) ≤ − ✏ if x / ∈ F , and 2. P y 6 = x Q ( x, y )( V ( y ) − V ( x )) ≤ K if x ∈ F , for some ✏ > 0 , K < ∞ and a bounded set F , then X ( t ) is positive recurrent.
Foster - Lyapunov Criteria 13 § The following Lyapunov Function will satisfy the Foster-Lyapunov criteria m m ⇣ ⌘ 2 ⇣ ⌘ + X X � � V ( x ) = ( ⇡ − ⇡ i ) | x | + C 1 (1 − ⇡ ) | x | + C 2 M − ⇡ i | x | | {z } i =1 i =1 L 2 | {z } | {z } L 1 L 3 § All are functions of state § We need to show that that the drift is negative except in some finite set
Intuition 14 § L 1 is the sum of differences in marginal m ⇣ ⌘ 2 X chunk frequencies. ) = ( ⇡ − ⇡ i ) | x | § Mode suppression increases but not ¯ i =1 π i π | {z } if π i < π L 1 L 2 decreases if increases. � � ¯ π + C 1 (1 − ⇡ ) | x | + In Mode Suppression this will happen only when | {z } the marginal chunk frequencies is uniform L 2 m ⇣ ⌘ + X } + C 2 | M − ⇡ i | x | § Number of chunks in the system increases i =1 | {z } L 3
Progression of Mode Suppression 15 1 2 3 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 1 D ( x ) = { 2 , 3 , 4 , 5 , 6 } D ( x ) = φ D ( x ) = { 5 } 5 4 1 2 3 4 5 6 1 2 3 4 5 6 D ( x ) = φ D ( x ) = { 1 , 2 , 4 , 5 , 6 }
Threshold Mode Suppression 16 § In MS any slight deviation from uniform marginal chunk frequency will result in suppression § Though this is favorable for stability, this will not result in best sojourn times § Is there a way to reduce the suppression of MS without compromising stability? § Use a “noisy” mode estimate: Threshold Mode Suppression(TMS) § The idea of TMS is to suppress the modes only if they are abundant compared to least frequency chunks § The set of indices suppressed in Threshold Mode Suppression are n o D T ( x ) = k | ⇡ k ( x ) = ⇡ ( x ) , ⇡ ( x ) | x | ≥ ⇡ ( x ) | x | + T
Threshold Mode Suppression 17 v Example when T=2 3 1 2 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 1 D ( x ) = { 2 , 3 , 4 , 5 , 6 } D ( x ) = φ D ( x ) = { 1 , 2 , 4 , 5 , 6 } D T ( x ) = { 2 , 3 , 4 , 5 , 6 } D T ( x ) = φ D T ( x ) = φ v When 𝑈 =1 , TMS = Mode Suppression v When 𝑈 →∞, TMS → Random Chunk because there won’t be any suppression
TMS - Stability 18 Theorem 2 The stability region of Threshold Mode Suppression (TMS) is λ > 0 for any finite threshold T < ∞ , if m ≥ 2 , µ > 0 and U > 0 . § Proof is using Foster-Lyapunov Criteria § Same Lyapunov function works § With the constant 𝐷↓ 1 >(2 𝑈 −1)( 𝑛 −1) C 1 m m ⇣ ⌘ 2 ⇣ ⌘ + X X � � V ( x ) = ( ⇡ − ⇡ i ) | x | + C 1 (1 − ⇡ ) | x | + C 2 M − ⇡ i | x | | {z } i =1 i =1 L 2 | {z } | {z } L 1 L 3
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